1. Field of the Invention
The present invention relates to computerized mathematical model for engineering purposes, and particularly to a method for Hammerstein modeling of a steam generator plant.
2. Description of the Related Art
The Hammerstein model belongs to a family of block-oriented models, and is made up of a memoryless nonlinear part followed by a linear dynamic part. It has been known to effectively represent and approximate several industrial processes, such as pH neutralization processes, distillation column processes, and heat exchange processes. Hammerstein models have also been used to successfully model nonlinear filters, biological systems, water heaters, and electrical drives.
A significant amount of research has been carried out on identification of Hammerstein models. Systems can be modeled by employing either nonparametric or parametric models. Nonparametric representations involve kernel regression or expansion of series, such as the Volterra series. This results in a theoretically infinite number of model parameters, and is therefore represented in terms of curves, such as step responses or bode diagrams. Parametric representations, such as state-space models, are more compact, as they have fewer parameters and the nonlinearity is expressed as a linear combination of finite and known functions.
Development of nonlinear models is the critical step in the application of nonlinear model based control strategies. Nonlinear behavior is prominent in the dynamic behavior of physical systems. Most physical devices have nonlinear characteristics outside a limited linear range. In most chemical processes, for example, understanding the nonlinear characteristics is important for designing controllers that regulate the process. It is rather difficult, yet necessary, to select a reasonable structure for the nonlinear model to capture the process nonlinearities. The nonlinear model used for control purposes should be as simple as possible, warranting minimal computational load and, at the same time, retaining most of the nonlinear dynamic characteristics of the system. The following convention has been used in what follows: upper case variables in bold represent matrices, lower case bold variables represent vectors, and lower case regular (i.e., non-bold) variables represent scalar quantities.
Many model structures have been proposed for the identification of nonlinear systems. The nonlinear static block followed by a dynamic block in the Hammerstein structure has been found to be a simple and effective representation for capturing the dynamics of typical chemical engineering processes such as distillation columns and heat exchangers, for example. Nonlinear system identification involves the following tasks: Structure selection, including selection of suitable nonlinear model structures and the number of model parameters; input sequence design, including the determination of the input sequence u(t) which is injected into the system to generate the output sequence y(t); noise modeling, which includes the determination of the dynamic model which generates the noise input w(t); parameter estimation, which includes estimation of the remaining model parameters from the dynamic system data u(t) and y(t), and the noise input w(t); and model validation, including the comparison of system data and model predictions for data not used in model development.
Hammerstein systems can be modeled by employing either nonparametric or parametric models. Nonparametric models represent the system in terms of curves resulting from expansion of series, such as the Volterra series or kernel regression. In practice, these curves are sampled, often leading to a large number of parameters. Parametric representations, such as state-space models, are more compact and have fewer parameters, while the nonlinearity is expressed as a linear combination of finite and known functions.
In the MIMO Hammerstein model, as noted above, a nonlinear system is represented as a nonlinear memory-less subsystem f(.), followed by a linear dynamic part. The input sequence u(t) and the output sequence y(t) are accessible to measurements, but the intermediate signal sequence v(t) is not. As shown in FIG. 1B, the static nonlinear element scales the inputs u(t) and transforms these inputs to v(t) through a nonlinear arbitrary function f(u). The dynamics of the system are modeled by a linear transfer function, whose outputs are y(t).
Many different techniques have been proposed for the black-box estimation of Hammerstein systems from input-output measurements. These techniques mainly differ in the way that static nonlinearity is represented and in the type of optimization problem that is finally obtained. In parametric approaches, the static nonlinearity is expressed in a finite number of parameters. Both iterative and non-iterative methods have been used for determination of the parameters of the static-nonlinear and linear-dynamic parts of the model. Typical techniques, however, are extremely costly in terms of computational time and energy.
Additionally, most techniques designed to deal with Hammerstein models focus purely on single-input single-output (SISO) models. Identification of MIMO systems, however, is a problem which has not been well explored. Identification based on prediction error methods (PEM), for example, is a complicated function of the system parameters, and has to be solved by iterative descent methods, which may get stuck into local minima. Further, optimization methods need an initial estimate for a canonical parametrization model; i.e. models with minimal numbers of parameters, which might not be easy to provide.
A typical steam generator plant may be identified as, or modeled with, a Hammerstein model. A typical steam generator plant is the Abbott Power Plant in Campaign, Ill. This plant is a dual fuel (oil/gas) fired unit used for heating and generating electric power. The plant has four inputs, namely fuel flow rate (scaled 0-1), air flow rate (scaled 0-1), water flow (inches), and steam demand disturbance (scaled 0-1), along with four outputs, namely drum pressure (psi), excess oxygen in exhaust gases (0-100%), water level in the drum (inches), and steam flow (kg/s). The plant is rated at 22.096 kg/s of steam at 22.4 MPa (325 psi) of pressure. The plant has dynamics of high order, as well as nonlinearities, instabilities, and time delays. FIG. 2 illustrates an overview of such a plant. In FIG. 2, u3 represents feed water flow, controlled by a suitable valve. Similarly, u1 represents fuel flow with a regulated rate, flowing across a burner within the furnace. y1 represents pressurized steam from a steam drum or the like with measured and regulated steam flow rate y4. The drum positioned beneath the steam drum may be a mud drum or the like. y2 represents measured excess oxygen, which is measured at the induced draft fan and a forced draft fan (the induced draft fan is shown as being above the forced draft fan in FIG. 2, though it should be understood that this is purely diagrammatic and representational). Further, u2 represents regulated and measured air flow and y3 represents the water level.
Apart from these measurable and deterministic outputs and inputs, there are certain disturbances in the plant, such as changes in steam demand by users and sensor noise, and certain uncertainties which include fuel calorific value variations, heat transfer coefficient variations, and distributed dynamics of steam generation. The plant also has few constraints, such as actuator constraints, unidirectional flow rates and drum flooding.
While models based on first principles and physical laws are available, a limited amount of work on steam and boiler modeling based on system identification exists. It would be desirable to be able to obtain a nonlinear model of steam generator plant directly from test data using the methods of system identification.
It has been shown that the minimal parametrization described above can lead to several problems. PEM have, therefore, inherent difficulties with MIMO system identification. More recent studies have also shown that maximum likelihood criterion results in a non-convex optimization problem in which global optimization is not guaranteed. Subspace identification methods (SIM) do not need nonlinear optimization techniques, nor do these methods need to impose to the system a canonical form. Subspace methods therefore do not suffer from the inconveniences encountered in applying PEM methods to MIMO system identification. Thus, it would be desirable to make use of this advantage, modeling the linear dynamic subsystem of the Hammerstein model with a state-space model rather than polynomial models. Thus, a method for Hammerstein modeling of a steam generator plant solving the aforementioned problems is desired.